
here, \(N\)  is the Poisson random measure with compensator \(\nu (\text{\rm {d}}z)\text{\rm {d}}t\) .
Set
\(\theta ^\lambda (x):=x+u^\lambda (x), x\in \mathbb {R}^d,\)  and
\(Z_t^{(\delta )}:=X_t-X^{(\delta )}_t\) . According to Itô's formula in [1]}, we obtain from
(REF ) that
\(\begin{split}\text{\rm {d}}\theta ^\lambda (X_t)&= \lambda u^\lambda (X_t)\text{\rm {d}}t+\text{\rm {d}}W_{S_t}+\int _{\mathbb {R}^d- \lbrace \bf 0\rbrace }[u^{\lambda }(X_{t-}+z)-u^\lambda (X_{t-})]\widetilde{N}(\text{\rm {d}}t,\text{\rm {d}}z)\\\text{\rm {d}}\theta ^\lambda (X^{(\delta )}_t)&=\lambda u^\lambda (X_t)\text{\rm {d}}t +\nabla \theta ^{\lambda }(X_t^{(\delta )})( b(X^{(\delta )}_{t_\delta })-b(X^{(\delta )}_t))\text{\rm {d}}t\\&+\text{\rm {d}}W_{S_t}+\int _{\mathbb {R}^d- \lbrace \bf 0\rbrace }[u^\lambda (X^{(\delta )}_{t-}+z)-u^\lambda (X^{(\delta )}_{t-})]\widetilde{N}(\text{\rm {d}}t, \text{\rm {d}}z).\end{split}\) 
